All it took was for Jay-Z, a member of US rap royalty, to flick a wad of €500 notes in a music video five years ago for people to wonder if the euro – visual shorthand for squareness – might one day become cool.

Would Andy Warhol's dollar screenprints give way to modern meditations on Europe's yellow € sign? Could the euro become more than a giant star-encrusted symbol outside Frankfurt's European Central Bank? Was it art? Europeans in the cultural world knew the answer was no.

1 January 2012 will mark 10 years since the euro coins and notes appeared in people's wallets. But as simple objects, pieces of design and branding, emotional items that bind us together, they are seen as a failure: a limp bureaucratic compromise where art was needed.

This was the first postmodern currency; it could have been visually extraordinary, combining the cutting-edge beauty of Dutch guilders and the design chic of the Swiss franc. Instead, the euro's visual blandness reflects its current identity crisis.

The euro is pure functionality in the extreme. So easy to pronounce, it has escaped the nicknames of its older cousins, the greenback and quid. (At the Maastricht summit, it was still called the ecu but Helmut Kohl thought it sounded too French – much to the relief of the Portuguese, as it also sounded like their word for arse).

The design was deliberately tepid. The notes feature neither people nor places, just bland, fake architecture that doesn't exist. Ten years ago the French economist André Orléan suspected this would become a problem: "Look at the symbolism: bridges and imaginary windows. The euro isn't anchored in the past, it's virtual, it doesn't correspond to any reality."

The French ethnologist Patrick Prado called it a "ghost money", with "no reference, no country, no past, no roots, no memory, defined by no value other than itself". He cautioned: "What will history make of this denial of images, this wiping out of the symbolic?"

The coins' national flipsides do give a nod to history, from Irish harps and Finnish swans to Leonardo da Vinci's beautiful Vitruvian Man on Italian cents. The French novelist Philippe Sollers has written about the joy of rummaging through loose change to find a coin bearing Cervantes that has travelled across borders into your pockets, with all the imaginary stories of how it got there.

In an ode to the euro in the Nouvel Observateur five years ago, he lauded the currency's silence: "The dollar bill is chatty, the euro is mute." Dollar bills shout: "In God We Trust." The staid euro would never dare quote Latin at you or suggest God was looking over your shoulder.

But Bruno Ninaber van Eyben, the designer who created the Dutch face of the euro coins, laments the "missed emotional opportunity". He hates the fact that the euro side of the coin shows a map and boundaries – and not even Poland and the members to the east.

"It shows borders, not what's inside them," he said. "That isn't what binds people together. There needs to be memory, emotional resonance, an idea of the future. The Romans understood this. When the coin bearing Augustus's face went all over Europe, there was a sense that people belonged to a group. They were not alone."

Erik Spiekermann, the German designer and typographer, adds: "It doesn't work as a brand. There's no intrinsic value in it. People aren't proud of it, they don't collect it. They just pay with it. It's like white bread, it won't harm you, but it doesn't nourish you either."

Is it a classic? "No. It would take another 20 years – if it lasts that long. You can't design a classic, something becomes a classic if it stays around. The dollar is a classic because of American cultural imperialism."

If half of the euro's neighbours, including many Balkan states, look up to it, others look down on it, not least the Swiss with their famously beautiful notes.

"It's ugly," says Pierre Frey, a Swiss art historian. "You have to look pretty hard to find such ugly inks. It's the image of the tempest that is currently shaking it."